Basics

Tests for nominal variables

Descriptive statistics

Tests for one measurement variable

Tests for multiple measurement variables

Multiple tests

Miscellany

Normality


Amphipod weight histogram
Histogram of dry weights of the amphipod crustacean Platorchestia platensis.
Amphipod weight histogram
Histogram of dry weights of the amphipod crustacean Platorchestia platensis.

One of the assumptions of an anova and other parametric tests is that the data are normally distributed. When you plot a frequency histogram, the frequencies should approximate the familiar bell-shaped normal curve. For example, the figure shown at the right is a histogram of dry weights of newly hatched amphipods (Platorchestia platensis). It fits the normal curve pretty well.


Two non-normal histograms
Two non-normal histograms.
Two non-normal histograms
Two non-normal histograms.

Other data sets don't fit the normal curve very well. The histogram on the top is the level of sulphate in Maryland streams (data from the Maryland Biological Stream Survey). It doesn't fit the normal curve very well, because there are a small number of streams with very high levels of sulphate. The histogram on the bottom is the number of egg masses laid by indivuduals of the lentago host race of the treehopper Enchenopa (unpublished data courtesy of Michael Cast). The curve is bimodal, with one peak at around 14 egg masses and the other at zero.

Like other parametric tests, the analysis of variance assumes that the data fit the normal distribution. If your measurement variable is not normally distributed, you may be increasing your chance of a false positive result if you analyze the data with an anova or other test that assumes normality. Fortunately, an anova is not very sensitive to moderate deviations from normality; simulation studies, using a variety of non-normal distributions, have shown that the false positive rate is not affected very much by this violation of the assumption (Glass et al. 1972, Harwell et al. 1992, Lix et al. 1996). This is because when you take a large number of random samples from a population, the means of those samples are approximately normally distributed even when the population is not normal.

It is possible to test the goodness-of-fit of a data set to the normal distribution. I do not suggest that you do this, because many data sets that are significantly non-normal would be perfectly appropriate for an anova.

Instead, if you have a large enough data set, I suggest you just look at the frequency histogram. If it looks more-or-less normal, go ahead and perform an anova. If it looks like a normal distribution that has been pushed to one side, like the sulphate data above, you should try different data transformations and see if any of them make the histogram look more normal. If that doesn't work, and the data still look severely non-normal, it's probably still okay to analyze the data using an anova. However, you may want to analyze it using a non-parametric test. Just about every parametric statistical test has a non-parametric substitute, such as the Kruskal–Wallis test instead of a one-way anova, Wilcoxon signed-rank test instead of a paired t-test, and Spearman rank correlation instead of linear regression. These non-parametric tests do not assume that the data fit the normal distribution. They do assume that the data in different groups have the same distribution as each other, however; if different groups have different shaped distributions (for example, one is skewed to the left, another is skewed to the right), a non-parametric test may not be any better than a parametric one.

If you want to check the normality of data for an anova, but you don't have enough observations in each group, you can look at the residuals of all of the observations. Subtract the mean of each group from each observation in that group, then look at the histogram of the combined set of all residuals. This won't tell you whether there are differences in the shape of the distribution among groups, but it is better than nothing.

Skewness and kurtosis

Five non-normal graphs
Graphs illustrating skewness and kurtosis.
Five non-normal graphs
Graphs illustrating skewness and kurtosis.

A histogram with a long tail on the right side, such as the sulphate data above, is said to be skewed to the right; a histogram with a long tail on the left side is said to be skewed to the left. There is a statistic to describe skewness, g1, but I don't know of any reason to calculate it; there is no rule of thumb that you shouldn't do an anova if g1 is greater than some cutoff value.

Another way in which data can deviate from the normal distribution is kurtosis. A histogram that has a high peak in the middle and long tails on either side is leptokurtic; a histogram with a broad, flat middle and short tails is platykurtic. The statistic to describe kurtosis is g2, but I can't think of any reason why you'd want to calculate it, either.

How to look at normality

Spreadsheet

I've written a spreadsheet that will plot a frequency histogram for untransformed, log-transformed and square-root transformed data. It will handle up to 1000 observations.

If there are not enough observations in each group to check normality, you may want to examine the residuals (each observation minus the mean of its group). To do this, open a separate spreadsheet and put the numbers from each group in a separate column. Then create columns with the mean of each group subtracted from each observation in its group, as shown below. Copy these numbers into the histogram spreadsheet.

Spreadsheet for residual calculation
A spreadsheet showing the calculation of residuals.
Spreadsheet for residual calculation
A spreadsheet showing the calculation of residuals.

Web pages

There are several web pages that will produce histograms, but most of them aren't very good. The interactive histogram web page is pretty cool. You enter your numbers (separated by spaces only, no tabs or line returns), and when you get a histogram, you can change the "binwidth" (the size of each interval on the histogram) by sliding a bar.

SAS

You can use the PLOTS option in PROC UNIVARIATE to get a stem-and-leaf display, which is a kind of very crude histogram. You can also use the HISTOGRAM option to get an actual histogram, but only if you know how to send the output to a graphics device driver.

Further reading

Sokal and Rohlf, pp. 698-703, 406-407.

Zar, pp. 185-188.

References

Glass, G.V., P.D. Peckham, and J.R. Sanders. 1972. Consequences of failure to meet assumptions underlying fixed effects analyses of variance and covariance. Rev. Educ. Res. 42: 237-288.

Harwell, M.R., E.N. Rubinstein, W.S. Hayes, and C.C. Olds. 1992. Summarizing Monte Carlo results in methodological research: the one- and two-factor fixed effects ANOVA cases. J. Educ. Stat. 17: 315-339.

Lix, L.M., J.C. Keselman, and H.J. Keselman. 1996. Consequences of assumption violations revisited: A quantitative review of alternatives to the one-way analysis of variance F test. Rev. Educ. Res. 66: 579-619.



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This page was last revised September 13, 2009. Its address is http://udel.edu/~mcdonald/statnormal.html. It may be cited as pp. 150-154 in: McDonald, J.H. 2009. Handbook of Biological Statistics (2nd ed.). Sparky House Publishing, Baltimore, Maryland.

©2009 by John H. McDonald. You can probably do what you want with this content; see the permissions page for details.